g(x) h(y) f(z) g(x) x h(y) y Ifqli/ECE305/problems/Chapter6_Problems.pdf · 280 CHAPTER 6...

274 CHAPTER 6 ELECTROSTATIC BOUNDARY-VALUE PROBLEMS

6.5 Which of the following is not true?

(a) -5cos3xisasolutionto<J>"(x) + 9</>(x) = 0

(b) 10 sin 2x is a solution to</>'' (x) - 4</>(x) = 0 (c) -4 cosh 3y is a solution to R" (y) - 9R(y) = 0

(d) sinh 2y is a solution to R" (y) - 4R(y) = 0

g" ( x) h" (y) (e) g(x) = - h(y) = f(z) = -1 where g(x) = sin x and h(y) = sinh y

6.6 If V1 = X1 Y1 is a product solution of Laplace's equation, which of these are not solutions of Laplace's equation?

(a) -10X1 Y1

(b) X 1Y1 + 2xy

(c) X 1Y1 - x + y

(d) X 1 + Y1

(e) (X1 - 2)(Y1 + 3)

6. 7 The capacitance of a capacitor filled by a linear dielectric is independent of the charge on the plates and the potential difference between the plates.

(a) True (b) False

6.8 A parallel-plate capacitor connected to a battery stores twice as much charge with a given dielectric as it does with air as dielectric. The susceptibility of the dielectric is

(a) 0

(b) 1

(c) 2

(d) 3

(e) 4

6.9 A potential difference V0

is applied to a mercury column in a cylindrical container. The mercury is now poured into another cylindrical container of half the radius and the same potential difference V0 applied across the ends. As a result of this change of space, the resistance will be increased

(a) 2 times

(b) 4 times

(c) 8 times

(d) 16 times

6.10 Two conducting plates are inclined at an angle 30° to each other with a point charge between them. The number of image charges is

(a) 12

(b) 11

(c) 6

(d) 5

(e) 3

Answers: 6.1 a, 6.2c, 6.3a, 6.4c, 6.Sb, 6.6d,e, 6.7 a, 6.8b, 6.9d, 6.1 Ob.

Section 6.2-Poisson's and Laplace's Equations

6.1 Given V = 5x3y2z and t: = 2.25e0 , find (a) Eat point P(-3, 1, 2), (b) Pv at P.

10 cos 0 sin <P 6.2 Let V = r2 and B = e0 • (a) Find Eat point P(l, 60°, 30°). (b) Determine Pv at P.

Problems 275

6.3 Conducting sheets are located at y = 1 and y = 3 planes. The space between them is

filled with a nonuniform charge distribution Pv = L_ nC/m3 and s = 4s0 • Assuming that 4'7T

V(y = 1) = 0 and V(y = 3) = 50 V, find V(y = 2).

6.4 Let V = sin 3

<f> V in a dielectric material for which s = 2.8 e0 • p

(a) Find E and P at point A(I, 20°, 4).

(b) Calculate the volume charge density at A.

6.5 A certain material occupies the space between two conducting slabs located at y = + 2 cm. When heated, the material emits electrons such that Pv = 50( 1 - y2

) µ,C/m3• If the slabs

are both held at 30 kV, find the potential distribution within the slabs. Takes = 3s0

•

6.6 Two large flat metal sheets are located at z = 0 and z = d and are maintained at 0 and V0 ,

respectively. The charge density between the sheets is Pv(z) = p;z/ d, where p0 is a constant. Determine the potential at all points between the plates.

6. 7 In cylindrical coordinates, V = 0 at p = 2 m and V = 60 V at p = 5 m due to charge 10

distribution Pv = - pC/m3• If Br= 3.6, find E.

p

6.8 Determine if each of the following potentials satisfies Laplace's equation.

(a) V1 = e-y sinh x

(b) V2 = cos x cos y cos z

(c) V3 = (Ap3 + Bp- 3) cos 3<f>

(d) V4

= 5 sine r2

6.9 Let U = 3xyz + y - i2. Show whether or not U satisfies Laplace's equation.

6.10 Given V = x3y + yz + cz2, find c such that V satifies Laplace's equation.

6.11 The potential field V = 2x2yz - y3z exists in a dielectric medium hayjng s = 2s0 •

(a) Does V satisfy Laplace's equation? (b) Calculate the total charge within the unit cube 0 < x < 1 m, 0 < y < 1 m, 0 < z < 1 m.

6.12 In cylindrical coordinates, V = 0 at p = 4 mm and V = Ya at p = 12 mm. If E = -6aP kV/mat p = 8 mm, determine V0 •

6.13 Consider the conducting plates shown in Figure 6.29. If V(z = 0) = 0 and V(z = 2 mm) = 50 V, determine V, E, and Din the dielectric region (er= 1.5) between the plates and Ps on the plates.

6.14 The cylindrical capacitor whose cross section is in Figure 6.30 has inner and outer radii of 5 mm and 15 mm, respectively. If V(p = 5 mm) = 100 V and V(p = 15 mm) = 0 V, calculate V, E, and D at p = 10 mm and Ps on each plate. Takes, = 2.0.

6.15 In cylindrical coordinates, V = 50 V on plane <f> = 71'/2 and V = 0 on plane <f> = 0. Assuming that the planes are insulated along the z-axis, determine E between the planes.


z

I + + + + + + +

d:;;; 2 mm 1 l l l l l lE \

V(z:;;; 0):;;; 0

FIGURE 6.29 For Problem 6.13.

IOOV

FIGURE 6.30 Cylindrical capacitor of Problem 6.14.

z

V = I 00 V '-....

FIGURE 6.31 Conducting cones of Problem 6 .20 .

*6.16 The potential field in a certain region is V = (c1p2 + c2p-2)sin 2¢. (a) Show that V satisfies

Laplace's equation. (b) Find c1 and c2 if V = 50 V and IEI = 100 V/m at point P(l, 45°, 1).

*6.17 (a) Show that V = V0 (1-a2 I p2) p sin <P (where V0 is constant) satisfies Laplace's equation.

(b) Determine E for p2 >> a2•

6. 18 Two conducting planes are located at x = 0 and x = 50 mm. The zero voltage reference is at x = 20 mm. Given that E = - 11 O(\; V /m, calculate the conductor voltages.

6. 19 The region between concentric spherical conducting shells r = 0.5 m and r = 1 m is charge free. If V(r = 0.5) = -50 V and V(r = 1) = 50 V, determine the potential distribution and the electric field strength in the region between the shells.

6.20 Find V and Eat (3, 0, 4) due to the two conducting cones of infinite extent shown in Figure 6.31.

*6.21 The inner and outer electrodes of a diode are coaxial cylinders of radii a = 0.6 mm and b = 30 mm, respectively. The inner electrode is maintained at 70 V, while the outer electrode is grounded. (a) Assuming that the length of the electrodes C >> a, band ignoring the effects of space charge, calculate the potential at p = 15 mm. (b) If an electron is injected radially through a small hole in the inner electrode with velocity 107 mis, find its velocity at p = 15 mm.

y

L__===========================-~i~--· x --

y y

Jr----/

·--v=o--- - V=O

I

Problems 277


V = V0

y

' a'-,--\--V=O-V = V/

0

/ _,.v _______ ,.,_ ___ x

o / b' ------..-.---x I ,,,_ ____________ ,__ __ x

/0 0 b b V = V0

(a) (b) (c)


6.22 An electrode with a hyperbolic shape (xy = 4) is placed above a grounded right-angle corner as in Figure 6.32. Calculate V and E at point ( 1, 2, 0) when the electrode is connected to a 20 V source.

*6.23 Solve Laplace's equation for the two-dimensional electrostatic systems of Figure 6.33 and find the potential V(x, y ).

*6.24 Find the potential V(x, y) due to the two-dimensional systems of Figure 6.34.

6.25 A conducting strip is defined as shown in Figure 6.34(b ). The potential distribution is

sin --(n7ry)

4V 00 a V(x,y ) = -

0 ~ exp( - n7Tx/ a) 7r n=odd n

Find the electric field E.

6.26 Figure 6.35 shows the cross-sectional view of an infinitely long rectangular slot. Find the potential distribution in the slot.


y

V=O

(a)

y

a

y

a

V=V o-.......___

(c)

FIGURE 6.34 For Problems 6.24 and 6.25.

y=O V=O

X a

X=a

. ;ry V= Vo sm -

b

Slot

....

y + x =O V=O

b


y = b

V=O

a

V= 0

(b)

6.27 By letting V(p, cf>) = R(p )<P( cf>) be the solution of Laplace's equation in a region where p * 0, show that the separated differential equations for R and <P are

R' A. R" + - - - R = 0

p p2

and <P" + A.<P = 0

where A is the separation constant.

Problems 279

FIGURE 6.36 For Problem 6.30 .

6.28 A potential in spherical coordinates is a function of r and () but not </>. Assuming that V(r, 0) = R(r)F(O), obtain the separated differential equations for Rand Fin a region for which Pv = 0.

Section 6.5-Resistance and Capacitance

6.29 Show that the resistance of the bar of Figure 6.17 between the vertical ends located at </> = 0 and </> = 7T' /2 is

7T' R= ---

b 2a-t ln -

a

*6.30 Show that the resistance of the sector of a spherical shell of conductivity <r, with cross section shown in Figure 6.36 (where 0 < <f> < 27T' ), between its base (i.e., from r = a to r = b) is

1 [ 1 I] R = 21T<r(l - cos a) a - b

6.31 A spherical shell has inner and outer radii a and b, respectively. Assume that the shell has a uniform conductivity <r and that it has copper electrodes plated on the inner and outer surfaces. Show that

1 (1 1) R = 41Tu a - b

*6.32 A hollow conducting hemisphere of radius a is buried with its flat face lying flush with the earth's surface, thereby serving as an earthing electrode. If the conductivity of earth is a-, show that the leakage conductance between the electrode and earth is 27T'aa-.


6.33 Another method of finding the capacitance of a capacitor is by using energy considerations, that is,

2WE 1 f C = - = - slEl2 dv v2 v2 0 0

Using this approach, derive eqs. (6.22), (6.28), and (6.32).

6.34 In an integrated circuit, a capacitor is formed by growing a silicon dioxide layer (sr = 4) of thickness 1 µ,m over the conducting silicon substrate and covering it with a metal electrode of area S. Determine S if a capacitance of 2 nF is desired.

6.35 Calculate the capacitance of the parallel-plate capacitor shown in Figure 6.37.

6.36 Evaluate the capacitance of the parallel-plate capacitor with multilayer dielectric shown in Figure 6.38.

6.37 The parallel-plate capacitor of Figure 6.39 is quarter-filled with mica (er = 6). Find the capacitance of the capacitor.

Depth= 15 cm

E,.3 = 8 2mm

20cm 20cm 20cm


e,2 = 5

Area= 80 cm2



I / 10 cm2

2mm

Problems 281


~x--

6.38 To appreciate the physical size of 1 F capacitor, consider a parallel-plate capacitor filled with air and with separation distance of 1 mm. Find the area of the plates to provide a capacitance of 1 F.

*6.39 An air-filled parallel plate capacitor of length L, width a, and plate separation d has its plates maintained at constant potential difference V 0 • If a dielectric slab of dielectric constant sr is slid between the plates and is withdrawn until only a length x remains between the plates as in Figure 6.40, show that the force tending to restore the slab to its original position is

8 0 (.sr - 1 ) a V~ F=------

2d

6.40 A parallel-plate capacitor has plate area 200 cm2 and plate separation of 3 mm. The charge density is 1 µ.,C/m2 and air is the dielectric. Find

(a) The capacitance of the capacitor

(b) The voltage between the plates

(c) The force with which the plates attract each other

6.41 The capacitance of a parallel-plate capacitor is 56 µF when the dielectric material is in place. The capacitance drops to 32 µF when the dielectric material is removed. Calculate the dielectric constant Br of the material.

6.42 A large parallel-plate capacitor has one plate at z = 0 and maintained at 40 V, while the other plate at z = 2 mm is maintained at 0 V. Find the potential between the plates.

6.43 A parallel-plate capacitor has a 4 mm plate separation, 0.5 m2 surface area per plate, and a dielectric with Er = 6.8. If the plates are maintained at 9 V potential difference, calculate

(a) the capacitance, (b) the charge density on each plate.

6.44 A parallel-plate capacitor with plate area S and spacing d has a charge Q on each plate. Assuming that the space between the plates is filled with dielectric Br= e re 0 , determine the energy stored when the plate spacing is (a) doubled, (b) halved.

6.45 A parallel-plate capacitor has separation 5 mm and area 0.4 m2• If the space between

the plates is filled with dielectric with er1 = 2.5, 0 < d < 1.5 mm, dielectric with er2 = 5.6, 1.5 mm < d < 3 mm, and dielectric with Br3 = 8.1, 3 mm< d < 5 mm, calculate the capacitance.

6.46 The space between spherical conducting shells r = 5 cm and r = 10 cm is filled with a dielectric material for which e = 2.25e0 • The two shells are maintained at a potential difference of 80 V. (a) Find the capacitance of the system. (b) Calculate the charge density on shell r = 5 cm.


y

a

0 a b x

FIGURE 6.41 For Problem 6.48. FIGURE 6.42 For Problem 6.50.

6.47 A spherical capacitor has inner radius d and outer radius a. Concentric with the spherical conductors and lying between them is a spherical shell of outer radius c and inner radius b. If the regions d < r < c, c < r < b, and b < r < a are filled with materials with permittivites 8 1, 8 2, and 8 3, respectively, determine the capacitance of the system.

6.48 Determine the capacitance of a conducting sphere surrounded by a thick spherical shell as shown in Figure 6.41.

6.49 A coaxial cable has inner radius of 5 mm and outer radius of 8 mm. If the cable is 3 km long, calculate its capacitance. Assume e = 2.5e0 •

6.50 A capacitor consists of two plates with equal width (b - a), and a length L in the z-direction. The plates are separated by <P = rr/4, as shown in Figure 6.42. Assume that the plates are separated by a dielectric material (e = e0er) and ignore fringing. Determine the capacitance.

6.51 Calculate the capacitance of a coaxial cable with inner radius 1 mm, outer radius 3.5 mm, and length 400 n1n1. Assu1ne the dielectric has e = 4.2e

0•

*6.52 In an ink-jet printer the drops are charged by surrounding the jet of radius 20 µ,m with a concentric cylinder of radius 600 µ.,mas in Figure 6.43. Calculate the minimum voltage required to generate a charge 50 fC on the drop if the length of the jet inside the cylinder is 100 µ,m. Take s = s 0 •

6.53 The cross section of a cable is shown in Figure 6.44. Determine the capacitance per unit length.

*6.54 A spherical capacitor has an inner conductor of radius a carrying charge Q and is maintained at zero potential. If the outer conductor contracts from a radius b to c under internal forces, prove that the work performed by the electric field as a result of the contraction is

Q2(b - c) W= ---

8rrsbc

Problems 283

+

Liquid ----"\ I ____ J

b

Liquid jet

FIGURE 6.43 Simplified geometry of an ink-jet printer; for Problem 6.52. FIGURE 6.44 For Problem 6.53.

*6.55 A parallel-plate capacitor has its plates at x = 0, d and the space between the plates is

filled with an inhomogeneous material with permittivity e = eo( 1 + ~). If the plate at

x = d is maintained at V0

while the plate at x = 0 is grounded, find:

(a) Vand E

(b) p

(c) P Ps at x = 0, d

( d) the capacitance, assuming that each plate has area S

6.56 Two parallel conducting plates are located at x = d and x = - d. The plate at x = d is held at V0 , while the plate at x = - d is grounded. If the space between the plates is filled with an inhomogeneous dielectric medium with

e= ----

1 + GY find the capacitance. Assume that each plate has an area S.

6.57 A spherical capacitor has inner radius a and outer radius b and is filled with an inhomogeneous dielectric withs = s 0 k/r. Show that the capacitance of the capacitor is

6.58 If the earth is regarded a spherical capacitor, what is its capacitance? Assume the radius of the earth to be approximately 6370 km.


6.59 A capacitor is formed by two coaxial metal cylinders of radii a = 1 mm and b = 5 mm. If the space between the cylinders is filled with a dielectric having er = 3 ( 1 + p), a< p < b, and pis in millimeters, determine the capacitance per meter.

Section 6.6-Method of Images

6.60 A grounded metal sheet is located in the z = 0 plane, while a point charge Q is located at (O, 0, a). Find the force acting on a point charge -Q placed at (a, 0, a).

6.61 Two point charges of 3 nC and -4 nC are placed, respectively, at ( 0, 0, 1 m ) and ( 0, 0, 2 m) while an infinite conducting plane is at z = 0. Determine

(a) The total charge induced on the plane

(b) The magnitude of the force of attraction between the charges and the plane

*6.62 A point charge of 10 µ,C is located at ( 1, 1, 1), and the positive portions of the coordinate planes are occupied by three mutually perpendicular plane conductors maintained at zero potential. Find the force on the charge due to the conductors.

6.63 A point charge Q is placed between two earthed intersecting conducting planes that are inclined at 45° to each other. Determine the number of image charges and their locations.

6.64 Infinite line x = 3, z = 4 carries 16 nC/m and is located in free space above the conducting plane z = 0. (a) Find Eat (2, -2, 3 ) . (b) Calculate the induced surface charge density on the conducting plane at ( 5, -6, 0).

6.65 In free space, infinite planes y = 4 and y = 8 carry charges 20 nC/m2 and 30 nC/m2,

respectively. If plane y = 2 is grounded, calculate Eat P( 0, 0, 0) and Q( -4, 6, 2 ).

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